This is Prince Eugene House, which even has its own Wikipedia page, which was constructed in 1817 on the former site of the Forforosa Gate. This location is significant as it marks the exact point where Prince Eugene of Savoy entered the city in 1716 after successfully seizing the fortress from Ottoman control, marking the start of Habsburg rule.
Although the house bears his name, the military commander never actually lived there, as construction began more than eighty years after his death. The building serves as a memorial to this transition of power, featuring a prominent medallion above the doorway of the above photo that depicts a basic representation of the original gate.
Over the centuries, the property has transitioned through various uses, serving as a Jewish house of prayer, a series of private commercial shops, a restaurant and pub (my favourite usage) before becoming a theatre agency during the mid-20th century. Following the period of nationalisation under the communist regime, the building was eventually returned to the descendants of the Savici family in the 1990s.
It’s perhaps in need of some repair today, although that’s true of a lot of the grand buildings in the city, although matters are clearly improving. The graffiti isn’t doing a great deal for the look of the building to be honest….